I've tried that. I've also restarted the SharePoint box.
It appears the chain of events was this:
1. An attempt to delete the UPS Application was made.
2. The deletion stayed in a stopping state. Additionally the service stayed in a stopping state.
3. Then, the Remove-SPServiceApplication was run.
4. The service application was removed but the sync service is still there, still in a "stopping" state.
I've attempted to run the Stop-SPServiceInstance command but that hasn't worked. Now, I can't run that command more than once. It throws an error stated that the service job instance already exists.....
I'm stuck at this point, having already restarted a multitude of services and the server itself.